\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  -1.0 & -10.00 x_{1} & -3.00 x_{2} & +  4.00 x_{3} & +  6.00 x_{4} & -1.00 x_{5}\\
 x_{7}   &  31.0 & -2.00 x_{1} & -9.00 x_{2} & +  3.00 x_{3} & -4.00 x_{4} & -4.00 x_{5}\\
 x_{8}   &  -15.0 & +  8.00 x_{1} & +  5.00 x_{2} & +  5.00 x_{3} & +  3.00 x_{4} & +  5.00 x_{5}\\
 x_{9}   &  29.0 & -4.00 x_{1} & -2.00 x_{2} & -5.00 x_{3} & +  1.00 x_{4} & -4.00 x_{5}\\
 x_{10}   &  26.0 & +  1.00 x_{1} & -2.00 x_{2} & -1.00 x_{3} &   & -8.00 x_{5}\\
 x_{11}   &  -3.0 & -1.00 x_{1} & +  6.00 x_{2} & +  5.00 x_{3} & + 10.00 x_{4} & -4.00 x_{5}\\
 x_{12}   &  18.0 & +  2.00 x_{1} & +  1.00 x_{2} & -4.00 x_{3} & +  2.00 x_{4} & -5.00 x_{5}\\
\hline
z    &  0.0 & +  4.00 x_{1} & -3.00 x_{2} & -5.00 x_{3} & -4.00 x_{4} & +  1.00 x_{5}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{5}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,12$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & + 10.00 y_{6} & +  2.00 y_{7} & -8.00 y_{8} & +  4.00 y_{9} & -1.00 y_{10} & +  1.00 y_{11} & -2.00 y_{12}\\
 y_{2}   &  1.0 & +  3.00 y_{6} & +  9.00 y_{7} & -5.00 y_{8} & +  2.00 y_{9} & +  2.00 y_{10} & -6.00 y_{11} & -1.00 y_{12}\\
 y_{3}   &  1.0 & -4.00 y_{6} & -3.00 y_{7} & -5.00 y_{8} & +  5.00 y_{9} & +  1.00 y_{10} & -5.00 y_{11} & +  4.00 y_{12}\\
 y_{4}   &  1.0 & -6.00 y_{6} & +  4.00 y_{7} & -3.00 y_{8} & -1.00 y_{9} &   & -10.00 y_{11} & -2.00 y_{12}\\
 y_{5}   &  1.0 & +  1.00 y_{6} & +  4.00 y_{7} & -5.00 y_{8} & +  4.00 y_{9} & +  8.00 y_{10} & +  4.00 y_{11} & +  5.00 y_{12}\\
\hline
z    &  -0 & +  1.00 y_{6} & -31.00 y_{7} & + 15.00 y_{8} & -29.00 y_{9} & -26.00 y_{10} & +  3.00 y_{11} & -18.00 y_{12}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 4 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{4}   &  0.268292682927 & -0.22 y_{1} & +  7.29 y_{7} & +  0.95 y_{3} & -4.88 y_{9} & -1.17 y_{10} & -5.02 y_{11} & -6.24 y_{12}\\
 y_{2}   &  0.256097560976 & +  0.43 y_{1} & +  9.10 y_{7} & +  0.32 y_{3} & -1.29 y_{9} & +  2.11 y_{10} & -4.84 y_{11} & -1.41 y_{12}\\
 y_{8}   &  0.170731707317 & -0.05 y_{1} & -0.27 y_{7} & -0.12 y_{3} & +  0.80 y_{9} & +  0.07 y_{10} & -0.56 y_{11} & +  0.39 y_{12}\\
 y_{6}   &  0.0365853658537 & +  0.06 y_{1} & -0.41 y_{7} & -0.10 y_{3} & +  0.24 y_{9} & +  0.16 y_{10} & -0.55 y_{11} & +  0.51 y_{12}\\
 y_{5}   &  0.182926829268 & +  0.30 y_{1} & +  4.93 y_{7} & +  0.51 y_{3} & +  0.22 y_{9} & +  7.79 y_{10} & +  6.26 y_{11} & +  3.56 y_{12}\\
\hline
z    &  2.59756097561 & -0.67 y_{1} & -35.44 y_{7} & -1.93 y_{3} & -16.68 y_{9} & -24.74 y_{10} & -5.96 y_{11} & -11.63 y_{12}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  0.670731707317 & +  0.22 x_{4} & -0.43 x_{2} & +  0.05 x_{8} & -0.06 x_{6} & -0.30 x_{5}\\
 x_{7}   &  35.4390243902 & -7.29 x_{4} & -9.10 x_{2} & +  0.27 x_{8} & +  0.41 x_{6} & -4.93 x_{5}\\
 x_{3}   &  1.92682926829 & -0.95 x_{4} & -0.32 x_{2} & +  0.12 x_{8} & +  0.10 x_{6} & -0.51 x_{5}\\
 x_{9}   &  16.6829268293 & +  4.88 x_{4} & +  1.29 x_{2} & -0.80 x_{8} & -0.24 x_{6} & -0.22 x_{5}\\
 x_{10}   &  24.743902439 & +  1.17 x_{4} & -2.11 x_{2} & -0.07 x_{8} & -0.16 x_{6} & -7.79 x_{5}\\
 x_{11}   &  5.96341463415 & +  5.02 x_{4} & +  4.84 x_{2} & +  0.56 x_{8} & +  0.55 x_{6} & -6.26 x_{5}\\
 x_{12}   &  11.6341463415 & +  6.24 x_{4} & +  1.41 x_{2} & -0.39 x_{8} & -0.51 x_{6} & -3.56 x_{5}\\
\hline
z    &  -2.59756097561 & -0.27 x_{4} & -0.26 x_{2} & -0.17 x_{8} & -0.04 x_{6} & -0.18 x_{5}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  0.670731707317 & +  0.22 x_{4} & -0.43 x_{2} & +  0.05 x_{8} & -0.06 x_{6} & -0.30 x_{5}\\
 x_{7}   &  35.4390243902 & -7.29 x_{4} & -9.10 x_{2} & +  0.27 x_{8} & +  0.41 x_{6} & -4.93 x_{5}\\
 x_{3}   &  1.92682926829 & -0.95 x_{4} & -0.32 x_{2} & +  0.12 x_{8} & +  0.10 x_{6} & -0.51 x_{5}\\
 x_{9}   &  16.6829268293 & +  4.88 x_{4} & +  1.29 x_{2} & -0.80 x_{8} & -0.24 x_{6} & -0.22 x_{5}\\
 x_{10}   &  24.743902439 & +  1.17 x_{4} & -2.11 x_{2} & -0.07 x_{8} & -0.16 x_{6} & -7.79 x_{5}\\
 x_{11}   &  5.96341463415 & +  5.02 x_{4} & +  4.84 x_{2} & +  0.56 x_{8} & +  0.55 x_{6} & -6.26 x_{5}\\
 x_{12}   &  11.6341463415 & +  6.24 x_{4} & +  1.41 x_{2} & -0.39 x_{8} & -0.51 x_{6} & -3.56 x_{5}\\
\hline
z    &  -6.9512195122 & +  1.63 x_{4} & -3.12 x_{2} & -0.41 x_{8} & -0.73 x_{6} & +  2.34 x_{5}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  0.670731707317 & +  0.22 x_{4} & -0.43 x_{2} & +  0.05 x_{8} & -0.06 x_{6} & -0.30 x_{5}\\
 x_{7}   &  35.4390243902 & -7.29 x_{4} & -9.10 x_{2} & +  0.27 x_{8} & +  0.41 x_{6} & -4.93 x_{5}\\
 x_{3}   &  1.92682926829 & -0.95 x_{4} & -0.32 x_{2} & +  0.12 x_{8} & +  0.10 x_{6} & -0.51 x_{5}\\
 x_{9}   &  16.6829268293 & +  4.88 x_{4} & +  1.29 x_{2} & -0.80 x_{8} & -0.24 x_{6} & -0.22 x_{5}\\
 x_{10}   &  24.743902439 & +  1.17 x_{4} & -2.11 x_{2} & -0.07 x_{8} & -0.16 x_{6} & -7.79 x_{5}\\
 x_{11}   &  5.96341463415 & +  5.02 x_{4} & +  4.84 x_{2} & +  0.56 x_{8} & +  0.55 x_{6} & -6.26 x_{5}\\
 x_{12}   &  11.6341463415 & +  6.24 x_{4} & +  1.41 x_{2} & -0.39 x_{8} & -0.51 x_{6} & -3.56 x_{5}\\
\hline
z    &  -6.9512195122 & +  1.63 x_{4} & -3.12 x_{2} & -0.41 x_{8} & -0.73 x_{6} & +  2.34 x_{5}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  1.11538461538 & -0.23 x_{3} & -0.50 x_{2} & +  0.08 x_{8} & -0.04 x_{6} & -0.42 x_{5}\\
 x_{7}   &  20.6666666667 & +  7.67 x_{3} & -6.67 x_{2} & -0.67 x_{8} & -0.33 x_{6} & -1.00 x_{5}\\
 x_{4}   &  2.02564102564 & -1.05 x_{3} & -0.33 x_{2} & +  0.13 x_{8} & +  0.10 x_{6} & -0.54 x_{5}\\
 x_{9}   &  26.5641025641 & -5.13 x_{3} & -0.33 x_{2} & -0.18 x_{8} & +  0.26 x_{6} & -2.85 x_{5}\\
 x_{10}   &  27.1153846154 & -1.23 x_{3} & -2.50 x_{2} & +  0.08 x_{8} & -0.04 x_{6} & -8.42 x_{5}\\
 x_{11}   &  16.141025641 & -5.28 x_{3} & +  3.17 x_{2} & +  1.21 x_{8} & +  1.06 x_{6} & -8.96 x_{5}\\
 x_{12}   &  24.2820512821 & -6.56 x_{3} & -0.67 x_{2} & +  0.41 x_{8} & +  0.13 x_{6} & -6.92 x_{5}\\
\hline
z    &  -3.64102564103 & -1.72 x_{3} & -3.67 x_{2} & -0.21 x_{8} & -0.56 x_{6} & +  1.46 x_{5}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  0.353361945637 & +  0.02 x_{3} & -0.65 x_{2} & +  0.02 x_{8} & -0.09 x_{6} & +  0.05 x_{11}\\
 x_{7}   &  18.8655221745 & +  8.26 x_{3} & -7.02 x_{2} & -0.80 x_{8} & -0.45 x_{6} & +  0.11 x_{11}\\
 x_{4}   &  1.05579399142 & -0.73 x_{3} & -0.52 x_{2} & +  0.06 x_{8} & +  0.04 x_{6} & +  0.06 x_{11}\\
 x_{9}   &  21.4377682403 & -3.45 x_{3} & -1.34 x_{2} & -0.56 x_{8} & -0.08 x_{6} & +  0.32 x_{11}\\
 x_{10}   &  11.9442060086 & +  3.73 x_{3} & -5.48 x_{2} & -1.06 x_{8} & -1.04 x_{6} & +  0.94 x_{11}\\
 x_{5}   &  1.80114449213 & -0.59 x_{3} & +  0.35 x_{2} & +  0.13 x_{8} & +  0.12 x_{6} & -0.11 x_{11}\\
 x_{12}   &  11.8125894134 & -2.48 x_{3} & -3.11 x_{2} & -0.52 x_{8} & -0.69 x_{6} & +  0.77 x_{11}\\
\hline
z    &  -1.00858369099 & -2.58 x_{3} & -3.15 x_{2} & -0.01 x_{8} & -0.39 x_{6} & -0.16 x_{11}\\
\end{array}\]
Final Dictionary
Solution:  -1.00858369099
Num Pivots:  3
\end{document}
